The air outside smells sickly sweet and I am dragged through the doors by my nostrils. The Enfield branch of Krispy Kreme doughnuts is cavernous, decorated in subway tiles and retro ice-cream colours, an attempt at recreating a bite of Americana, instead of a large shack in a retail park just off the A10. The music piped through the building is Elvis Presley, a man who knew a thing or two about doughnuts.

This week, the North Carolina-based chain announced it would be doubling the number of its outlets in the UK to 100 over the next five years, on the back of growing sales, with revenue up 12.6% and serving 5 million customers. The next will open in Leeds next month.

What is more astonishing is the way each new store is received. When Krispy Kreme opened its most recent outlet in Cardiff in April, its 46th in the UK, more than 1,000 people queued for two hours for a doughnut. One couple even camped overnight at the front of the queue. "I do have a life, honestly," the 32-year-old Londoner told a Welsh newspaper, before going on to explain that it's his hobby to go to the openings of every new Krispy Kreme store. He will be busy over the next few years.

The company has a wide and ever-changing range of doughnuts – lemon meringue pie, chocolate dreamcake and mango passion cheesecake are all current flavours – and are displayed in glass cabinets like baubles in a jewellery shop.

Here in the UK, they are stocked in Harrods and Selfridges, a marketing masterstroke, because there is also a less glamorous retail side. You find kiosks in motorway service stations, and in Enfield – the UK's first stand-alone store after it had opened a concession in Harrods to much hype – Krispy Kreme shares its car park with a McDonald's, where people sit in their fetid cars, masticating. This store even has a drive-through, for when you really can't be bothered to walk the few metres to the counter.

The brand is infused with a glamorous American mythology, helped in part by cameo appearances in television shows such as Sex and the City and The Sopranos, with well-publicised fans such as Madonna and Beyoncé. It is an example of great branding, says Don Williams, chief executive of brand consultancy PI Global. "It has a strong personality, its heritage is almost tangible, and yet it doesn't feel like it's had marketeers all over it. It looks like it's always been that way – a little bit of old Americana, in the same way that Coca-Cola has its roots in that world. Notice the logo similarities."

But why are they doing so well in the UK at a time when the anti-obesity message seems to be everywhere? "There are two massive trends in food right now – one is health, the other is indulgence, and both are similar in size," says Rob Ward, a food marketing expert. "It's a bit of a myth that we're all charging off to the salad aisle; we're equally charging off to indulge ourselves in treat food. One of the symptoms of the state of economic uncertainty is to go for comfort food that gives you instant gratification."

Fat and sugar – the twin pillars of the best comfort food. The company's classic glazed doughnut contains 217 calories and 13g fat; the Cookies & Kreme doughnut has 380 calories and 17g fat. In 2008, the company announced it was removing trans fats, the toxic processed fat that clogs arteries, but 0.5g remains (US food labelling allows companies to "round down", which is why the company sometimes claims "zero trans fats"). "It is something that we're continuing to work on," says Judith Denby, the UK's marketing manager of Krispy Kreme.

"We are running a trial of doughnuts with no trans fats in a couple of our stores, and if it goes well we will be rolling that out." It isn't even the doughnuts which are its worst offenders – Krispy Kreme also make drinks, and its lemon sherbet chiller was widely claimed to have an incredible 980 calories and 40g of fat. (Denby hasn't heard of it: "I presume it was a US thing.")

Of course, nobody can possibly be unaware that doughnuts are not a healthy snack. "They are definitely a treat item, not something to have regularly," says Sian Porter, consultant dietitian and spokesperson for the British Dietetic Association. "Also, you have to look at the way they sell them – they offer discounts for a box, so there is the temptation to buy more than you want, which can encourage people to eat more."

I was lured to the Enfield store by the Krispy Kreme website. "Come along and join in the party games fun with us," it promised in its list of store events, "to entertain the kids this half-term". Party games? Fried foodstuffs? The warm glow of self-righteous indignation at the way they market calorific treats to children? What a way to spend the afternoon. It's a disappointment. There is no sign of any games and I only spot one child. Two men in work shirts sit by the window, eating doughnuts from a large box on the table in front of them. When they leave, a group of teenage girls take their place.

At one end of the store is a glass wall, through which you can watch the small factory at work as the doughnuts are made. Gobbets of dough swell and are then gently tipped into hot oil. They bob merrily along for a couple of minutes before being collected by a conveyor belt. A sign above it proclaims this the "Krispy Kreme Theatre". It is interesting enough, but nothing much happens – Waiting for Godough, if you like.

Actually, says Rob Ward, this is an essential part of the brand. "Another of the big trends in food is trust, because there has been a massive breakdown. What you are seeing is a demonstration of the product being made, and when you see that, you trust it more."

Meanwhile, the recipes are a closely guarded secret – and I don't notice any leaflets with nutritional information in the store. Also, the corporate emphasis on freshness means unsold doughnuts are always thrown away (it would be "irresponsible" to give them away, says Denby, because they are "a treat and not a meal".)

The staff behind the counter are so friendly I am coaxed into buying two doughnuts in a special "deal" when I only meant to buy one. "You can take one home," says the woman behind the counter, with a smile as sweet as butterscotch. "One thing that does drive me mad about Krispy Kreme," says Ward, "is, when I go with my kids, the way they try to encourage you to buy a dozen, when I only want two. The 'supersize' element, as a parent, is annoying, and is something other companies, such as McDonald's, moved away from."

I am baffled by the choice, but settle on a Strawberries & Kreme, a doughnut with a jammy, creamy centre. It is toothsome, but I can only eat half before it makes me feel a bit sick.

There is no doubt that other people find these doughnuts irresistible. I have a colleague who would drive for an hour to this very store, "At midnight. Several times a month. Then I'd eat a dozen – yes, a dozen – and have horrible sugar crashes." She had, she explains with understatement, "a slight addiction". A visit to the company's British Facebook page reveals people begging them to open stores in their areas, with feverish desperation.

Outside the Enfield store, two teenage girls carry their three-dozen doughnuts for a birthday party. "Nobody else does doughnuts like these," says 18-year-old Amma Osei-Owusu. A man in a shirt and tie carries a box to his car. "It's a treat for a customer," says Chris, a sales manager for a textiles firm. Ah, the sweet taste of bribery. He laughs. "I call it corporate entertainment. I wouldn't normally spend so much on a doughnut for myself. They're not cheap." Starting at £1.20 for a plain doughnut, they are more than twice the price of your average British version.

Even in the absence of "party games fun", business seems brisk, but the recent history of Krispy Kreme should be, say some market commentators, a business school case study in how to almost ruin your brand. In the 1930s, Vernon Rudolph bought a recipe from a New Orleans French pastry chef for yeast doughnuts, and in 1937 he started to sell them to local grocery stores in Winston-Salem, North Carolina.

The sugary aroma of his factory drew customers in, so he cut a hole in the wall to serve them and cut out the middlemen. In the 50s, there were several stores, and Rudolph had opened a factory to make doughnut mix to deliver to them; today the company makes a significant proportion of its money from selling huge bags of the mix to its franchises (in the UK, all stores are company-owned, but in the US, most are franchises). By the end of the 90s, the company had gone national, and in 2000 it was floated on the stock exchange. Over the next couple of years, the company expanded rapidly, including internationally. They spent little or no money on advertising, relying instead on word of mouth and giveaways. In the weeks before it opened its Harrods concession, the company would regularly send boxes of doughnuts to the office of the newspaper where I worked. This is one of its ingrained marketing tricks. Another is its fundraising. The company allows people to buy boxes of doughnuts at cost price, and then sell them for a profit for a good cause, and many of these are schools or children's groups such as Scouts and Guides. It's a genius idea – raise money for charity, while getting kids to sell to other kids, and mint a whole new generation of parent-pestering doughnut-eaters. "I ordered 120 doughnuts for a sale at school and they all went within 40 mins," reads one testimonial on the British website, "I'm definitely doing it again!"

"This is a marketing company," one shareholder, Erskine Bowles, the North Carolina banker and Bill Clinton's former chief of staff, told Fortune magazine in 2003. "Scott Livengood [the then CEO] once told me if a finance guy ever gets hold of Krispy Kreme, sell every share." In 2003, shares hit almost $50.

Then it started to go wrong. In 2004, the company's first loss was blamed on the rise of the low-carbohydrate Atkins diet, but it was much more than this. There was a litany of bad business decisions, and some questionable accounting led to Livengood being investigated by America's federal securities and exchange commission. Stores were closed, franchisees filed for bankruptcy and sales continued to fall.

The price of expansion is ubiquity – Krispy Kreme doughnuts were now sold in supermarkets and petrol stations – which had slightly taken the sheen off their once rare and exclusive image. The glass cabinet showcasing them in my local Tesco, for instance, was meant to conjure up some of the glossiness of the brand, but it soon started to look a bit shabby; it disappeared some months later. In 2009, the share price hit a historic low of just over $1.

Today, the company is in better shape. In the last year in the UK alone, it has opened six new stores and posted sales of £33.4m, compared to £29.7m in the previous year. Good news for the company, not so great news for anti-obesity campaigners, particularly those who are concerned about the way it popularises itself with children – all week it has been promoting half-term face-painting and doughnut-decorating activities in its stores.

"The government should regulate to protect children from the marketing of these junk foods, as voluntary approaches are clearly not working," says Charlie Powell, campaigns director of the Children's Food Campaign. "The fight against childhood obesity goes on, in spite of the best efforts of food companies like Krispy Kreme." The response from Krispy Kreme is as sugary as its doughnuts. "It's about an occasional treat," says Judith Denby. "We believe everyone deserves a treat every now and then."

Sweet victory: Four steps to doughnut domination

On screen

The brand made its film debut in 1998's Primary Colours when Governor Jack Stanton, played by John Travolta, sat in a Krispy Kreme diner chewing the fat with a waiter. Subsequently, Krispy Kremes have made many on-screen appearances, being regularly consumed in the Boston legal offices of 90s comedy Ally McBeal. But perhaps the best-remembered appearance of the doughnut was in series five of Sex and the City, when Miranda escapes the rigours of a Weight Watchers diet by sharing a Krispy Kreme with her overweight date.

The royal connection?

Apparently consumed across generations of the royal family, the company saw fit to release a commemorative golden ring doughnut ahead of this year's royal wedding. Ever the doting aunt, Princess Anne was reportedly spotted purchasing a box of 12 just before the big day. Meanwhile, upon her return to the UK after a gap year in 2009, Princess Eugenie attended an exclusive party with pole dancers, a lavish dinner and piles of Krispy Kremes. Neither report has been confirmed by Buckingham Palace, indicating the company may be benefiting from a shrewd PR team rather than royal approval.

Celebrity endorsement

The list of high-profile fans is long: Jonathan Ross tweets pictures of his; Beyoncé has said they're one of her major cravings; Jimmy Carr rewards audiences at select gigs with them after his shows; and Cheryl Cole sent Black Eyed Peas frontman will.i.am a box full when his album hit No 1. No wonder gossip magazines have got in on the scene, with Glamour creating their own range of sparkly, lip gloss-inspired doughnuts earlier this year.

Politics

Playing its part in the drive to get Americans to the polling booths during the 2008 presidential elections, Krispy Kreme outlets rewarded those who had cast their ballot with a free doughnut. To celebrate Barack Obama's victory, the chain offered punters a free coffee with its very own piece of coffee art – an image of Obama in the foam – rebranding the drink "The United States of Americano". Oliver Laughland